In general, horizontal axis fluid turbine rotor blades comprise two to five blades arranged evenly about a central axis and coupled to an electrical generation machine.
Generally speaking, a fluid turbine structure with an open unshrouded rotor design captures energy from a fluid stream that is smaller in diameter than the rotor. In an open unshrouded rotor fluid turbine, as fluid flows from the upstream side of the rotor to the downstream side, the average axial fluid velocity remains constant as the flow passes through the rotor plane. Energy is extracted at the rotor resulting in a pressure drop on the downstream side of the rotor. The fluid directly downstream of the rotor is at sub-atmospheric pressure due to the energy extraction and the fluid directly upstream of the rotor is at greater than atmospheric pressure. The high pressure upstream of the rotor deflects some of the upstream air around the rotor. In other words, a portion of the fluid stream is diverted around the open rotor as if by an impediment. As the fluid stream is diverted around the open rotor, it expands, which is referred to as flow expansion at the rotor. Due to the flow expansion, the upstream area of the fluid flow is smaller than the area of the rotor.
The Betz limit calculates the maximum power that can be extracted from a volume of moving fluid by an open blade, horizontal axial flow turbine (HAWT). The Betz limit is derived from fluid dynamic control-volume theory for flow passing through an open rotor, and by applying one-dimensional equations based on the principles of the conservation of mass, momentum and energy. According to the Betz limit, and independent of the design of the fluid turbine, a maximum of 16/27 of the total kinetic energy in a volume of moving fluid can be captured by an open-rotor turbine. Conventional turbines commonly produce 75% to 80% of the Betz limit.
A fluid turbine power coefficient (Cp) is the power generated over the ideal power available by extracting all the wind kinetic energy approaching the rotor area. The Betz power coefficient of 16/27 is the maximum power generation possible based on the kinetic energy of the flow approaching the rotor, and the rotor area. For an open rotor, the rotor area used in the Betz Cp derivation is the system maximum flow area described by the diameter of the rotor blades. The maximum power generation occurs when the rotor flow velocity is the average of the upstream and downstream velocity. This is the only rotor velocity that allows the flow-field to be reversible, and the power extraction to be maximized. At this operating point, the rotor velocity is ⅔ the wind velocity, the wake velocity is ⅓ the wind velocity, and the rotor flow has a non-dimensional pressure coefficient of −⅓ at the rotor exit. The −⅓ pressure coefficient is a result of the rotor wake flow expanding out to twice the rotor exit area downstream of the rotor station.
Induced drag is generated by a rotor blade due to the redirection of fluid during the generation of lift as a column of fluid flows through the rotor plane. The redirection of the fluid may include span-wise flow along the pressure side of the rotor blade along a radial direction toward the blade tip where the fluid then flows over to the opposite side of the blade. The fluid flow over the tips joins a chord-wise flow, otherwise referred to as bypass flow, forming rotor-tip vortices. The rotor-tip vortices mix with vortices shed from the trailing edge of the rotor blade to form the rotor wake.
It is commonly known that the rotor wake affects the rotor intake. A column of fluid encounters a rotor as an impediment, in part, because a portion of the fluid flowing around the rotor expands in the wake of the rotor in a form referred to as the stream column. Fluid flowing around the rotor plane is referred to as the bypass flow. Bypass flow passes over the outer surface of the stream column. Since the stream column can be considered to be comprised of an infinite fore-body and an infinite after-body, the resulting pressure force on the stream column is zero (refer to D'Alemberts paradox). Increasing lift over the rotor, and hence increasing the amount of energy extracted from the stream column, creates slower moving flow in the rotor wake, therefore, impeding flow through the rotor. This impediment increases the volume of the rotor wake. In other words, as more power is extracted at the rotor, the rotor stream column will expand and more fluid flow will bypass the rotor. As a result, maximum power is achieved from the two opposing effects of: increased power extraction resulting in relatively lower flow rates; and reduced power extraction resulting in relatively higher flow rates.
Proposed solutions to the above mentioned paradox include: increasing the size of the wake area to allow for increased wake expansion; and injecting high-energy fluid into the rotor wake. Both solutions have been proven to allow for increased energy extraction at the rotor.
Using idealized but broadly representative models, the power coefficient of a dual-tip rotor blade based upon rotor diameter is increased over a non-tip rotor blade by the ratio of the velocity at the location of the rotor blade, divided by the free-stream fluid velocity. This is measured as velocity (U) at the rotor blade plane (P) at a power extraction factor of zero (0), referred to as UP-0. Similarly, a rotor extracting power is measured as velocity (U) at the rotor blade plane (P) from minimum power production, up to a rated power (R), referred to as UP-R.
A fluid power coefficient (Cp) is a function of wake velocity ratio and thrust coefficient (Ct). Thrust coefficient is the ratio of pressure drop across the rotor over the dynamic pressure of the wind flow, which represents a critical parameter for the design of the rotor.
Providing an area for wake expansion in the down-stream region of the rotor plane results in a low exit-plane pressure coefficient (CTE) that allows for a relatively higher rotor-thrust coefficient.